Stability Matters for Reaction–Diffusion–Equations on Metric Graphs Under the Anti-Kirchhoff Vertex Condition

Abstract

The stability properties of stationary nonconstant solutions of reaction–diffusion–equations $\partial _t u_j=\partial _j^2u_{j}+f(u_j)$ on the edges k _j of a finite metric graph G under the so–called anti–Kirchhoff condition (KC) at the vertices v _i of the graph are investigated. The latter one consists in the following two requirements at each node.

$$\displaystyle \sum _{v_i\in k_j} u_{j}(v_i,t)=0, $$

$$\displaystyle k_j\cap k_s =\{v_i\}\;\Longrightarrow \; d_{ij}\partial _ju_{j}(v_i,t) =d_{is}\partial _s u_{s}(v_i,t), $$

where d _ij∂ _ju _j(v _i, t) denotes the outer normal derivative of u _j at v _i on the edge k _j. Though on any finite metric graph there is a simple nonlinearity leading to a unique stable nonconstant stationary solution, there are classes of reaction-terms allowing only stable stationary solutions that are constant on each edge. For example, odd nonlinearities allow only such stable stationary solutions, in particular they only allow the trivial solution as stable one on trees.

The second author was supported by the MINECO grant MTM2017-84214-C2-1-P and is part of the Catalan research group 2017 SGR 1392.